Body impulse from strength. Impulse conservation law. Where did the term "impulse" come from?
Body impulse
The impulse of a body is a quantity equal to the product of the mass of the body by its speed.
It should be remembered that we are talking about a body, which can be represented as a material point. The momentum of the body ($ p $) is also called the amount of motion. The concept of momentum was introduced into physics by René Descartes (1596-1650). The term "impulse" appeared later (impulsus means "push" in Latin). Impulse is a vector quantity (like speed) and is expressed by the formula:
$ p↖ (→) = mυ↖ (→) $
The direction of the impulse vector always coincides with the direction of the velocity.
The unit of impulse in SI is the impulse of a body with a mass of $ 1 $ kg, moving with a velocity of $ 1 $ m / s, therefore, the unit of an impulse is $ 1 $ kg $ · $ m / s.
If a constant force acts on the body (material point) during the time interval $ ∆t $, then the acceleration will also be constant:
$ a↖ (→) = ((υ_2) ↖ (→) - (υ_1) ↖ (→)) / (∆t) $
where, $ (υ_1) ↖ (→) $ and $ (υ_2) ↖ (→) $ are the initial and final velocities of the body. Substituting this value into the expression of Newton's second law, we get:
$ (m ((υ_2) ↖ (→) - (υ_1) ↖ (→))) / (∆t) = F↖ (→) $
Opening the brackets and using the expression for the momentum of the body, we have:
$ (p_2) ↖ (→) - (p_1) ↖ (→) = F↖ (→) ∆t $
Here $ (p_2) ↖ (→) - (p_1) ↖ (→) = ∆p↖ (→) $ is the change in momentum during the time $ ∆t $. Then the previous equation will take the form:
$ ∆p↖ (→) = F↖ (→) ∆t $
The expression $ ∆p↖ (→) = F↖ (→) ∆t $ is a mathematical representation of Newton's second law.
The product of force by the time of its action is called impulse of power... So the change in the momentum of a point is equal to the change in the momentum of the force acting on it.
The expression $ ∆p↖ (→) = F↖ (→) ∆t $ is called body motion equation... It should be noted that one and the same action - a change in the momentum of a point - can be obtained with a small force in a long period of time and with a large force in a short period of time.
The impulse of the tel. Impulse change law
The momentum (momentum) of a mechanical system is a vector equal to the sum of the impulses of all material points of this system:
$ (p_ (system)) ↖ (→) = (p_1) ↖ (→) + (p_2) ↖ (→) + ... $
The laws of change and conservation of momentum are a consequence of Newton's second and third laws.
Consider a system consisting of two bodies. The forces ($ F_ (12) $ and $ F_ (21) $ in the figure, with which the bodies of the system interact with each other, are called internal.
Let, in addition to internal forces, external forces $ (F_1) ↖ (→) $ and $ (F_2) ↖ (→) $ act on the system. For each body, we can write the equation $ ∆p↖ (→) = F↖ (→) ∆t $. Adding the left and right sides of these equations, we get:
$ (∆p_1) ↖ (→) + (∆p_2) ↖ (→) = ((F_ (12)) ↖ (→) + (F_ (21)) ↖ (→) + (F_1) ↖ (→) + (F_2) ↖ (→)) ∆t $
According to Newton's third law, $ (F_ (12)) ↖ (→) = - (F_ (21)) ↖ (→) $.
Hence,
$ (∆p_1) ↖ (→) + (∆p_2) ↖ (→) = ((F_1) ↖ (→) + (F_2) ↖ (→)) ∆t $
On the left side there is a geometric sum of changes in the impulses of all bodies of the system, equal to the change in the momentum of the system itself - $ (∆p_ (system)) ↖ (→) $. Taking this into account, the equality $ (∆p_1) ↖ (→) + (∆p_2) ↖ (→) = ((F_1) ↖ (→) + (F_2) ↖ (→)) ∆t $ can be written:
$ (∆p_ (system)) ↖ (→) = F↖ (→) ∆t $
where $ F↖ (→) $ is the sum of all external forces acting on the body. The result obtained means that the momentum of the system can only be changed by external forces, and the change in the momentum of the system is directed in the same way as the total external force. This is the essence of the law of change in the momentum of a mechanical system.
Internal forces cannot change the total impulse of the system. They only change the impulses of the individual bodies of the system.
Momentum conservation law
The law of conservation of momentum follows from the equation $ (∆p_ (sist)) ↖ (→) = F↖ (→) ∆t $. If no external forces act on the system, then the right-hand side of the equation $ (∆p_ (system)) ↖ (→) = F↖ (→) ∆t $ vanishes, which means that the total impulse of the system remains unchanged:
$ (∆p_ (system)) ↖ (→) = m_1 (υ_1) ↖ (→) + m_2 (υ_2) ↖ (→) = const $
A system that is not acted upon by any external forces or the resultant external forces is zero is called closed.
The momentum conservation law states:
The total impulse of a closed system of bodies remains constant for any interactions of the bodies of the system with each other.
The result obtained is valid for a system containing an arbitrary number of bodies. If the sum of external forces is not equal to zero, but the sum of their projections to some direction is equal to zero, then the projection of the momentum of the system on this direction does not change. So, for example, a system of bodies on the surface of the Earth cannot be considered closed due to the force of gravity acting on all bodies, however, the sum of the projections of impulses on the horizontal direction can remain unchanged (in the absence of friction), since in this direction the force of gravity does not acts.
Jet propulsion
Let us consider examples confirming the validity of the law of conservation of momentum.
Let's take a child rubber ball, inflate it and let it go. We will see that when the air starts to leave it in one direction, the ball itself will fly in the other. Ball movement is an example of jet propulsion. It is explained by the law of conservation of momentum: the total momentum of the "ball plus air in it" system before the air outflow is equal to zero; it must remain equal to zero during movement; therefore, the ball moves in the direction opposite to the direction of the outflow of the jet, and with such a speed that its momentum is equal in magnitude to the momentum of the air jet.
Reactive motion is called the movement of a body that occurs when some part of it separates from it at any speed. Due to the law of conservation of momentum, the direction of motion of the body is opposite to the direction of motion of the separated part.
Rocket flights are based on the principle of jet propulsion. The modern space rocket is a very complex aircraft. The mass of the rocket consists of the mass of the propellant (that is, the incandescent gases formed as a result of fuel combustion and emitted in the form of a jet stream) and the final, or, as they say, the "dry" mass of the rocket remaining after the ejection of the propellant from the rocket.
When a jet gas jet is ejected from a rocket at high speed, the rocket itself rushes in the opposite direction. According to the law of conservation of momentum, the momentum $ m_ (p) υ_p $ acquired by the rocket must be equal to the momentum $ m_ (gas) υ_ (gas) $ of the ejected gases:
$ m_ (p) υ_p = m_ (gas) υ_ (gas) $
Hence it follows that the rocket speed
$ υ_p = ((m_ (gas)) / (m_p)) υ_ (gas) $
It can be seen from this formula that the speed of the rocket is the greater, the greater the speed of the emitted gases and the ratio of the mass of the working body (ie, the mass of the fuel) to the final ("dry") mass of the rocket.
The formula $ υ_p = ((m_ (gas)) / (m_p)) υ_ (gas) $ is approximate. It does not take into account that as the fuel burns, the mass of the rocket in flight becomes less and less. The exact formula for the rocket speed was obtained in 1897 by K.E. Tsiolkovsky and bears his name.
Work of force
The term "work" was introduced into physics in 1826 by the French scientist J. Poncelet. If in everyday life only human labor is called work, then in physics and, in particular, in mechanics, it is generally accepted that work is done by force. The physical quantity of work is usually denoted by the letter $ A $.
Work of force Is a measure of the action of a force, depending on its modulus and direction, as well as on the movement of the point of application of the force. For constant force and linear movement, the work is determined by the equality:
$ A = F | ∆r↖ (→) | cosα $
where $ F $ is the force acting on the body, $ ∆r↖ (→) $ is the displacement, $ α $ is the angle between the force and the displacement.
The work of force is equal to the product of the moduli of force and displacement and the cosine of the angle between them, that is, to the scalar product of the vectors $ F↖ (→) $ and $ ∆r↖ (→) $.
Work is a scalar quantity. If $ α 0 $, and if $ 90 °
When several forces act on the body, the total work (the sum of the work of all forces) is equal to the work of the resulting force.
The unit of work in SI is joule($ 1 $ J). $ 1 $ J is the work that a $ 1 $ N force does on the way to $ 1 $ m in the direction of the action of this force. This unit is named after the English scientist J. Joule (1818-1889): $ 1 $ J = $ 1 $ N $ · $ m. Kilojoules and millijoules are also often used: $ 1 $ kJ $ = 1,000 $ J, $ 1 $ mJ $ = 0.001 $ J.
Work of gravity
Consider a body sliding along an inclined plane with an inclination angle $ α $ and a height $ H $.
Let us express $ ∆x $ in terms of $ H $ and $ α $:
$ ∆x = (H) / (sinα) $
Taking into account that the gravity force $ F_t = mg $ makes an angle ($ 90 ° - α $) with the direction of movement, using the formula $ ∆x = (H) / (sin) α $, we obtain an expression for the work of the gravity force $ A_g $:
$ A_g = mg · cos (90 ° -α) · (H) / (sinα) = mgH $
It can be seen from this formula that the work of gravity depends on the height and does not depend on the angle of inclination of the plane.
It follows that:
- the work of gravity does not depend on the shape of the trajectory along which the body moves, but only on the initial and final position of the body;
- when a body moves along a closed trajectory, the work of gravity is zero, that is, gravity is a conservative force (forces that have this property are called conservative).
Reaction forces work, is equal to zero, since the reaction force ($ N $) is directed perpendicular to the displacement of $ ∆x $.
Frictional force work
The friction force is directed opposite to the displacement of $ ∆x $ and makes an angle with it $ 180 ° $, therefore, the work of the friction force is negative:
$ A_ (tr) = F_ (tr) ∆x cos180 ° = -F_ (tr) ∆x $
Since $ F_ (tr) = μN, N = mgcosα, ∆x = l = (H) / (sinα), $ then
$ A_ (tr) = μmgHctgα $
Elastic force work
Let an external force $ F↖ (→) $ act on an unstretched spring of length $ l_0 $, stretching it by $ ∆l_0 = x_0 $. In position $ x = x_0F_ (control) = kx_0 $. After the cessation of the action of the force $ F↖ (→) $ at the point $ х_0 $, the spring is compressed under the action of the force $ F_ (control) $.
Let us determine the work of the elastic force when the coordinate of the right end of the spring changes from $ x_0 $ to $ x $. Since the elastic force in this section changes linearly, in Hooke's law, you can use its average value in this section:
$ F_ (ctrl.) = (Kx_0 + kx) / (2) = (k) / (2) (x_0 + x) $
Then the work (taking into account that the directions $ (F_ (cf. compare)) ↖ (→) $ and $ (∆x) ↖ (→) $ coincide) is equal to:
$ A_ (control) = (k) / (2) (x_0 + x) (x_0-x) = (kx_0 ^ 2) / (2) - (kx ^ 2) / (2) $
It can be shown that the form of the last formula does not depend on the angle between $ (F_ (cf. compare)) ↖ (→) $ and $ (∆x) ↖ (→) $. The work of the elastic forces depends only on the deformations of the spring in the initial and final states.
Thus, elastic force, like gravity, is a conservative force.
Power of force
Power is a physical quantity measured by the ratio of work to the period of time during which it is produced.
In other words, power shows how much work is done per unit of time (in SI - for $ 1 $ s).
Power is determined by the formula:
where $ N $ is the power, $ A $ is the work done in the time $ ∆t $.
Substituting into the formula $ N = (A) / (∆t) $ instead of work $ A $ its expression $ A = F | (∆r) ↖ (→) | cosα $, we get:
$ N = (F | (∆r) ↖ (→) | cosα) / (∆t) = Fυcosα $
Power is equal to the product of the moduli of the force and velocity vectors by the cosine of the angle between these vectors.
SI power is measured in watts (W). One watt ($ 1 $ W) is such a power at which $ 1 $ J work is done for $ 1 $ s: $ 1 $ W $ = 1 $ J / s.
This unit is named after the English inventor J. Watt (Watt), who built the first steam engine. J. Watt (1736-1819) himself used another unit of power - horsepower (hp), which he introduced in order to be able to compare the performance of a steam engine and a horse: $ 1 hp. $ = 735.5 $ W.
In technology, larger units of power are often used - kilowatts and megawatts: $ 1 $ kW $ = $ 1000 W, $ 1 $ MW $ = $ 1,000,000 W.
Kinetic energy. The law of change in kinetic energy
If a body or several interacting bodies (a system of bodies) can do work, then they say that they have energy.
The word "energy" (from the Greek energia - action, activity) is often used in everyday life. So, for example, people who can quickly do work are called energetic, having great energy.
The energy that a body possesses due to motion is called kinetic energy.
As in the case of the definition of energy in general, we can say about kinetic energy that kinetic energy is the ability of a moving body to do work.
Let us find the kinetic energy of a body with mass $ m $, moving with velocity $ υ $. Since kinetic energy is energy due to motion, the zero state for it is the state in which the body is at rest. Having found the work necessary to impart a given speed to the body, we will find its kinetic energy.
To do this, we calculate the work on the section of displacement $ ∆r↖ (→) $ when the directions of the force vectors $ F↖ (→) $ and displacement $ ∆r↖ (→) $ coincide. In this case, the work is equal to
where $ ∆x = ∆r $
For the movement of a point with acceleration $ α = const $, the expression for the movement has the form:
$ ∆x = υ_1t + (at ^ 2) / (2), $
where $ υ_1 $ is the initial speed.
Substituting into the equation $ A = F ∆x $ the expression for $ ∆x $ from $ ∆x = υ_1t + (at ^ 2) / (2) $ and using Newton's second law $ F = ma $, we get:
$ A = ma (υ_1t + (at ^ 2) / (2)) = (mat) / (2) (2υ_1 + at) $
Expressing the acceleration in terms of the initial $ υ_1 $ and final $ υ_2 $ velocities $ a = (υ_2-υ_1) / (t) $ and substituting in $ A = ma (υ_1t + (at ^ 2) / (2)) = (mat) / (2) (2υ_1 + at) $ we have:
$ A = (m (υ_2-υ_1)) / (2) (2υ_1 + υ_2-υ_1) $
$ A = (mυ_2 ^ 2) / (2) - (mυ_1 ^ 2) / (2) $
Now equating the initial speed to zero: $ υ_1 = 0 $, we obtain an expression for kinetic energy:
$ E_K = (mυ) / (2) = (p ^ 2) / (2m) $
Thus, a moving body has kinetic energy. This energy is equal to the work that needs to be done to increase the speed of the body from zero to the value of $ υ $.
From $ E_K = (mυ) / (2) = (p ^ 2) / (2m) $ it follows that the work of the force to move the body from one position to another is equal to the change in kinetic energy:
$ A = E_ (K_2) -E_ (K_1) = ∆E_K $
Equality $ A = E_ (K_2) -E_ (K_1) = ∆E_K $ expresses the theorem on the change in kinetic energy.
Change in the kinetic energy of the body(material point) for a certain period of time is equal to the work done during this time by the force acting on the body.
Potential energy
Potential energy is the energy determined by the mutual arrangement of interacting bodies or parts of the same body.
Since energy is defined as the body's ability to do work, then potential energy is naturally defined as the work of a force that depends only on mutual disposition Tel. This is the work of gravity $ A = mgh_1-mgh_2 = mgH $ and the work of elastic force:
$ A = (kx_0 ^ 2) / (2) - (kx ^ 2) / (2) $
The potential energy of the body, interacting with the Earth, is called a quantity equal to the product of the mass $ m $ of this body by the acceleration of gravity $ g $ and by the height $ h $ of the body above the surface of the Earth:
The potential energy of an elastically deformed body is a value equal to half the product of the elasticity (stiffness) coefficient $ k $ of the body and the square of the deformation $ ∆l $:
$ E_p = (1) / (2) k∆l ^ 2 $
The work of conservative forces (gravity and elasticity), taking into account $ E_p = mgh $ and $ E_p = (1) / (2) k∆l ^ 2 $, is expressed as follows:
$ A = E_ (p_1) -E_ (p_2) = - (E_ (p_2) -E_ (p_1)) = - ∆E_p $
This formula allows you to give a general definition of potential energy.
The potential energy of a system is a quantity depending on the position of bodies, the change in which during the transition of the system from the initial state to the final state is equal to the work of the internal conservative forces of the system, taken with the opposite sign.
The minus sign on the right-hand side of the equation $ A = E_ (p_1) -E_ (p_2) = - (E_ (p_2) -E_ (p_1)) = - ∆E_p $ means that when doing work by internal forces (for example, falling body on the ground under the action of gravity in the "stone - Earth" system), the energy of the system decreases. Work and change in potential energy in the system always have opposite signs.
Since work determines only a change in potential energy, then only a change in energy has a physical meaning in mechanics. Therefore, the choice of the zero energy level is arbitrary and is determined solely by considerations of convenience, for example, the simplicity of writing the corresponding equations.
The law of change and conservation of mechanical energy
Full mechanical energy of the system the sum of its kinetic and potential energies is called:
It is determined by the position of the bodies (potential energy) and their speed (kinetic energy).
According to the kinetic energy theorem,
$ E_k-E_ (k_1) = A_p + A_ (pr), $
where $ A_p $ is the work of potential forces, $ A_ (pr) $ is the work of non-potential forces.
In turn, the work of potential forces is equal to the difference in the potential energy of the body in the initial $ E_ (p_1) $ and final $ E_p $ states. With this in mind, we obtain an expression for the law of change in mechanical energy:
$ (E_k + E_p) - (E_ (k_1) + E_ (p_1)) = A_ (pr) $
where the left side of equality is the change in the total mechanical energy, and the right side is the work of non-potential forces.
So, mechanical energy change law reads:
The change in the mechanical energy of the system is equal to the work of all non-potential forces.
A mechanical system in which only potential forces operate is called conservative.
In the conservative system, $ A_ (pr) = 0 $. this implies mechanical energy conservation law:
In a closed conservative system, the total mechanical energy is conserved (does not change over time):
$ E_k + E_p = E_ (k_1) + E_ (p_1) $
The law of conservation of mechanical energy is derived from Newton's laws of mechanics, which are applicable to a system of material points (or macroparticles).
However, the law of conservation of mechanical energy is also valid for a system of microparticles, where Newton's laws themselves no longer apply.
The law of conservation of mechanical energy is a consequence of the homogeneity of time.
Time uniformity consists in the fact that under the same initial conditions, the course of physical processes does not depend on the moment at which these conditions are created.
The law of conservation of total mechanical energy means that with a change in kinetic energy in a conservative system, its potential energy should also change, so that their sum remains constant. This means the possibility of converting one type of energy into another.
In accordance with various forms of motion of matter, various types of energy are considered: mechanical, internal (equal to the sum of the kinetic energy of the chaotic movement of molecules relative to the center of mass of the body and the potential energy of interaction of molecules with each other), electromagnetic, chemical (which consists of the kinetic energy of the movement of electrons and electrical the energies of their interaction with each other and with atomic nuclei), nuclear, etc. From what has been said it is clear that the division of energy into different types is rather arbitrary.
Natural phenomena are usually accompanied by the transformation of one type of energy into another. So, for example, friction of parts of various mechanisms leads to the transformation of mechanical energy into heat, that is, into internal energy. In heat engines, on the contrary, there is a transformation of internal energy into mechanical energy; in galvanic cells, chemical energy is converted into electrical energy, etc.
Currently, the concept of energy is one of the basic concepts of physics. This concept is inextricably linked with the idea of the transformation of one form of movement into another.
This is how the concept of energy is formulated in modern physics:
Energy is a general quantitative measure of the movement and interaction of all types of matter. Energy does not arise from nothing and does not disappear, it can only pass from one form to another. The concept of energy connects together all natural phenomena.
Simple mechanisms. Efficiency of mechanisms
Simple mechanisms are called devices that change the magnitude or direction of forces applied to the body.
They are used to move or lift large loads with little effort. These include the lever and its varieties - blocks (movable and fixed), gate, inclined plane and its varieties - wedge, screw, etc.
Lever arm. Leverage rule
The arm is a solid body that can rotate around a fixed support.
The leverage rule says:
A lever is in balance if the forces applied to it are inversely proportional to their shoulders:
$ (F_2) / (F_1) = (l_1) / (l_2) $
From the formula $ (F_2) / (F_1) = (l_1) / (l_2) $, applying the property of proportion to it (the product of the extreme terms of the proportion is equal to the product of its middle terms), you can get the following formula:
But $ F_1l_1 = M_1 $ is the moment of force tending to turn the lever clockwise, and $ F_2l_2 = M_2 $ is the moment of force tending to turn the lever counterclockwise. Thus, $ M_1 = M_2 $, as required.
The lever began to be used by people in ancient times. With its help, it was possible to lift heavy stone slabs when building pyramids in Ancient egypt... Without leverage, this would not have been possible. Indeed, for example, for the construction of the Cheops pyramid, which has a height of $ 147 m, more than two million boulders were used, the smallest of which had a mass of $ 2.5 $ tons!
Nowadays, levers are widely used both in production (for example, cranes) and in everyday life (scissors, wire cutters, scales).
Fixed block
The action of a fixed block is similar to the action of a lever with equal arms: $ l_1 = l_2 = r $. The applied force $ F_1 $ is equal to the load $ F_2 $, and the equilibrium condition is:
Fixed block used when it is necessary to change the direction of the force without changing its magnitude.
Movable block
The movable block acts like a lever, the arms of which are: $ l_2 = (l_1) / (2) = r $. In this case, the equilibrium condition has the form:
where $ F_1 $ is the applied force, $ F_2 $ is the load. The use of a movable block gives a twofold gain in strength.
Polyspast (block system)
A normal pulley block consists of $ n $ movable and $ n $ fixed blocks. Its application gives a gain in strength in $ 2n $ times:
$ F_1 = (F_2) / (2n) $
Power pulley consists of n movable and one fixed block. The use of a power-law pulley block gives a gain in strength by $ 2 ^ n $ times:
$ F_1 = (F_2) / (2 ^ n) $
Screw
The screw is an inclined plane wound on an axis.
The equilibrium condition for the forces acting on the propeller has the form:
$ F_1 = (F_2h) / (2πr) = F_2tgα, F_1 = (F_2h) / (2πR) $
where $ F_1 $ - external force applied to the screw and acting at a distance $ R $ from its axis; $ F_2 $ - force acting in the direction of the screw axis; $ h $ - screw pitch; $ r $ - average radius of the thread; $ α $ - angle of inclination of the thread. $ R $ is the length of the arm (wrench) that rotates the screw with a force of $ F_1 $.
Efficiency
Coefficient of performance (COP) - the ratio of useful work to all spent work.
Efficiency is often expressed as a percentage and is denoted by the Greek letter $ η $ ("this"):
$ η = (A_п) / (A_3) 100% $
where $ A_n $ is useful work, $ A_3 $ is all the work spent.
Useful work is always only a part of the total work that a person spends using this or that mechanism.
Part of the perfect work is spent overcoming frictional forces. Since $ A_3> A_n $, the efficiency is always less than $ 1 $ (or $< 100%$).
Since each of the works in this equality can be expressed in the form of the product of the corresponding force and the distance traveled, it can be rewritten as follows: $ F_1s_1≈F_2s_2 $.
It follows that, winning with the help of a mechanism in force, we lose the same number of times on the way, and vice versa... This law is called the golden rule of mechanics.
The golden rule of mechanics is an approximate law, since it does not take into account the work to overcome the friction and gravity of the parts of the devices used. Nevertheless, it can be very useful in analyzing the operation of any simple mechanism.
So, for example, thanks to this rule, we can immediately say that the worker shown in the figure, with a two-fold gain in lifting power by $ 10 $ cm, will have to lower the opposite end of the lever by $ 20 $ cm.
Collision of bodies. Elastic and inelastic shock
The laws of conservation of momentum and mechanical energy are used to solve the problem of the motion of bodies after a collision: the values of these quantities after the collision are determined from the known impulses and energies before the collision. Consider the cases of elastic and inelastic shocks.
A blow is called absolutely inelastic, after which the bodies form a single body moving at a certain speed. The problem of the velocity of the latter is solved using the law of conservation of momentum for a system of bodies with masses $ m_1 $ and $ m_2 $ (if we are talking about two bodies) before and after impact:
$ m_1 (υ_1) ↖ (→) + m_2 (υ_2) ↖ (→) = (m_1 + m_2) υ↖ (→) $
Obviously, the kinetic energy of bodies during an inelastic impact is not conserved (for example, for $ (υ_1) ↖ (→) = - (υ_2) ↖ (→) $ and $ m_1 = m_2 $ it becomes zero after the impact).
A shock is called absolutely elastic, in which not only the sum of impulses is preserved, but also the sum of the kinetic energies of the impacting bodies.
For an absolutely elastic impact, the equations
$ m_1 (υ_1) ↖ (→) + m_2 (υ_2) ↖ (→) = m_1 (υ "_1) ↖ (→) + m_2 (υ" _2) ↖ (→); $
$ (m_ (1) υ_1 ^ 2) / (2) + (m_ (2) υ_2 ^ 2) / (2) = (m_1 (υ "_1) ^ 2) / (2) + (m_2 (υ" _2 ) ^ 2) / (2) $
where $ m_1, m_2 $ are the masses of the balls, $ υ_1, υ_2 $ are the velocities of the balls before impact, $ υ "_1, υ" _2 $ are the velocities of the balls after impact.
Themes of the USE codifier: momentum of a body, momentum of a system of bodies, law of conservation of momentum.
Pulse body is a vector quantity equal to the product of the body's mass by its velocity:
There are no special units of measure for impulse. The dimension of momentum is simply the product of the dimension of mass and the dimension of velocity:
Why is the concept of momentum interesting? It turns out that it can be used to give Newton's second law a slightly different, also extremely useful form.
Newton's second law in impulse form
Let be the resultant of the forces applied to the body of mass. We start with the usual writing of Newton's second law:
Taking into account that the acceleration of the body is equal to the derivative of the velocity vector, Newton's second law is rewritten as follows:
We introduce a constant under the derivative sign:
As you can see, the derivative of the impulse is obtained on the left side:
. ( 1 )
Relation (1) is a new form of writing Newton's second law.
Newton's second law in impulse form. The derivative of the momentum of the body is the resultant of the forces applied to the body.
You can also say this: the resulting force acting on the body is equal to the rate of change in the body's momentum.
The derivative in formula (1) can be replaced by the ratio of final increments:
. ( 2 )
In this case, there is an average force acting on the body during the time interval. The smaller the value, the closer the ratio is to the derivative, and the closer the average force is to its instantaneous value at a given moment in time.
In tasks, as a rule, the time interval is rather short. For example, it can be the time the ball hits the wall, and then the average force acting on the ball from the side of the wall during the strike.
The vector on the left-hand side of relation (2) is called change of momentum during . The change in momentum is the difference between the final and initial vectors of the momentum. Namely, if is the momentum of the body at some initial moment of time, is the momentum of the body after a period of time, then the change in momentum is the difference:
We emphasize again that the change in momentum is the difference of vectors (Fig. 1):
For example, let the ball fly perpendicular to the wall (the impulse before the impact is equal) and bounces back without losing speed (the impulse after the impact is equal). Despite the fact that the modulus of the impulse has not changed (), there is a change in the impulse:
Geometrically, this situation is shown in Fig. 2:
The modulus of the impulse change, as we can see, is equal to the doubled modulus of the initial impulse of the ball:.
Let's rewrite formula (2) as follows:
, ( 3 )
or, describing the change in momentum, as above:
The quantity is called impulse of power. There is no special unit of measure for the impulse of force; the dimension of the impulse of force is simply the product of the dimensions of force and time:
(Note that turns out to be another possible unit of measure for body momentum.)
The verbal formulation of equality (3) is as follows: the change in the momentum of the body is equal to the momentum of the force acting on the body for a given period of time. This, of course, is again Newton's second law in impulse form.
Force calculation example
As an example of applying Newton's second law in impulse form, let's consider the following problem.
Task.
A ball of mass g, flying horizontally at a speed of m / s, hits a smooth vertical wall and bounces off it without losing speed. The angle of incidence of the ball (that is, the angle between the direction of movement of the ball and the perpendicular to the wall) is equal to. Strike lasts for. Find the average strength,
acting on the ball during impact.
Solution. Let us show first of all that the angle of reflection is equal to the angle of incidence, that is, the ball will bounce off the wall at the same angle (Fig. 3).
According to (3) we have:. Hence it follows that the vector of change in momentum co-directional with a vector, that is, directed perpendicular to the wall in the direction of the ball's rebound (Fig. 5).
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Vectors and
equal in modulus
(since the speed of the ball has not changed). Therefore, a triangle composed of vectors and is isosceles. This means that the angle between the vectors and is equal, that is, the angle of reflection is really equal to the angle of incidence.
Now note, in addition, that our isosceles triangle has an angle (this is the angle of incidence); therefore, this triangle is equilateral. Hence:
And then the required average force acting on the ball:
The impulse of the system of bodies
Let's start with a simple situation for a two-body system. Namely, let there be body 1 and body 2 with impulses and, respectively. The momentum of the system of these bodies is the vector sum of the impulses of each body:
It turns out that for the momentum of a system of bodies there is a formula similar to Newton's second law in the form (1). Let's deduce this formula.
All other objects with which the bodies 1 and 2 we are considering interact, we will call external bodies. The forces with which external bodies act on bodies 1 and 2 are called external forces. Let be the resulting external force acting on body 1. Similarly, the resulting external force acting on body 2 (Fig. 6).
In addition, bodies 1 and 2 can interact with each other. Let body 2 act on body 1 with force. Then body 1 acts on body 2 with force. According to Newton's third law, the forces and are equal in magnitude and opposite in direction:. Forces and is internal forces, operating in the system.
Let's write for each body 1 and 2 Newton's second law in the form (1):
, ( 4 )
. ( 5 )
Let us add equalities (4) and (5):
On the left side of the obtained equality is the sum of derivatives, equal to the derivative of the sum of vectors and. On the right side, we have by virtue of Newton's third law:
But - this is the impulse of the system of bodies 1 and 2. Let's also designate - this is the resultant of external forces acting on the system. We get:
. ( 6 )
In this way, the rate of change of the momentum of a system of bodies is the resultant of external forces applied to the system. Equality (6), which plays the role of Newton's second law for a system of bodies, is what we wanted to obtain.
Formula (6) was derived for the case of two bodies. Now let us generalize our reasoning to the case of an arbitrary number of bodies in the system.
The impulse of the system of bodies bodies is called the vector sum of the impulses of all bodies included in the system. If the system consists of bodies, then the momentum of this system is:
Then everything is done in exactly the same way as above (only technically it looks a little more complicated). If for each body we write down equalities similar to (4) and (5), and then add all these equalities, then on the left side we again get the derivative of the impulse of the system, and on the right side there will be only the sum of external forces (internal forces, adding in pairs, will give zero in view of Newton's third law). Therefore, equality (6) remains valid in the general case.
Momentum conservation law
The system of bodies is called closed, if the actions of external bodies on the bodies of a given system are either negligibly small or cancel each other out. Thus, in the case of a closed system of bodies, only the interaction of these bodies with each other, but not with any other bodies, is essential.
The resultant of external forces applied to the closed system is zero:. In this case, from (6) we obtain:
But if the derivative of the vector vanishes (the rate of change of the vector is zero), then the vector itself does not change with time:
Impulse conservation law. The momentum of a closed system of bodies remains constant over time for any interactions of bodies within this system.
The simplest problems on the law of conservation of momentum are solved according to the standard scheme, which we will now show.
Task. A body of mass g moves at a speed of m / s on a smooth horizontal surface. A body of mass r is moving towards it with a speed of m / s. An absolutely inelastic shock occurs (the bodies stick together). Find the speed of bodies after impact.
Solution. The situation is shown in Fig. 7. The axis is directed towards the movement of the first body.
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Since the surface is smooth, there is no friction. Since the surface is horizontal and movement occurs along it, the force of gravity and the reaction of the support balance each other:
Thus, the vector sum of the forces applied to the system of these bodies is equal to zero. This means that the system of bodies is closed. Therefore, the law of conservation of momentum is fulfilled for it:
. ( 7 )
The impulse of the system before the impact is the sum of the impulses of the bodies:
After an inelastic impact, one body of mass was obtained, which moves with the required speed:
From the law of conservation of momentum (7) we have:
From here we find the speed of the body formed after the impact:
Let's move on to the projections on the axis:
By condition, we have: m / s, m / s, so that
The minus sign indicates that the stuck together bodies move in the direction opposite to the axis. Seeking speed: m / s.
Impulse projection conservation law
The following situation is often encountered in tasks. The system of bodies is not closed (the vector sum of external forces acting on the system is not zero), but there is such an axis, the sum of the projections of external forces on the axis is zero at any given time. Then we can say that along a given axis, our system of bodies behaves like a closed one, and the projection of the momentum of the system onto the axis is preserved.
Let us show this more rigorously. Let's project equality (6) onto the axis:
If the projection of the resultant external forces vanishes, then
Therefore, the projection is a constant:
Impulse projection conservation law. If the projection onto the axis of the sum of external forces acting on the system is zero, then the projection of the momentum of the system does not change over time.
Let's look at an example of a specific problem, how the law of conservation of momentum projection works.
Task. A mass boy, skating on smooth ice, throws a mass stone at an angle to the horizon. Find the speed with which the boy rolls back after being thrown.
Solution. The situation is shown schematically in Fig. eight . The boy is depicted as a straightforward.
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The impulse of the "boy + stone" system is not stored. This can be seen at least from the fact that after the throw, the vertical component of the impulse of the system appears (namely, the vertical component of the impulse of the stone), which was not there before the throw.
Therefore, the system formed by the boy and the stone is not closed. Why? The fact is that the vector sum of the external forces is not equal to zero during the throw. The value is greater than the sum, and due to this excess, the vertical component of the momentum of the system appears.
However, external forces act only vertically (no friction). Therefore, the projection of the momentum on the horizontal axis is preserved. Before the throw, this projection was zero. Directing the axis towards the throw (so that the boy went in the direction of the negative semiaxis), we get.
V Everyday life in order to characterize a person who commits spontaneous actions, the epithet "impulsive" is sometimes used. At the same time, some people do not even remember, and a significant part do not even know at all with what physical quantity this word is associated. What is hidden under the concept of "body impulse" and what properties does it possess? Such great scientists as René Descartes and Isaac Newton were looking for answers to these questions.
Like any science, physics operates with clearly formulated concepts. At the moment, the following definition has been adopted for a quantity called the impulse of a body: it is a vector quantity, which is a measure (amount) of the mechanical movement of a body.
Suppose that the issue is considered within the framework of classical mechanics, i.e., it is believed that the body moves with ordinary, and not with relativistic speed, which means that it is at least an order of magnitude less than the speed of light in vacuum. Then the pulse module of the body is calculated using formula 1 (see photo below).
Thus, by definition, this value is equal to the product of the body's mass by its velocity, with which its vector is co-directed.
In SI (International System of Units), 1 kg / m / s is taken as the unit of measurement for impulse.
Where did the term "impulse" come from?
Several centuries before the concept of the amount of mechanical motion of a body appeared in physics, it was believed that the cause of any movement in space is a special force - impetus.
In the 14th century, Jean Buridan made adjustments to this concept. He suggested that the flying cobblestone has an impetus directly proportional to its speed, which would be unchanged if there were no air resistance. At the same time, according to this philosopher, bodies with greater weight had the ability to "contain" more of such a driving force.
Further development of the concept, later called impulse, was given by Rene Descartes, who designated it with the words "momentum". However, he did not take into account that speed has a direction. That is why the theory put forward by him in some cases contradicted experience and did not find recognition.
The English scientist John Wallis was the first to guess that the momentum should also have a direction. It happened in 1668. However, it took another couple of years for him to formulate the well-known law of conservation of momentum. The theoretical proof of this fact, established empirically, was given by Isaac Newton, who used the third and second laws of classical mechanics discovered by him and named after him.
The momentum of the system of material points
Let us first consider the case when we are talking about speeds much lower than the speed of light. Then, according to the laws of classical mechanics, the total momentum of a system of material points is a vector quantity. It is equal to the sum of the products of their masses at speed (see formula 2 in the picture above).
In this case, the momentum of one material point is taken as a vector quantity (formula 3), which is codirectional with the speed of the particle.
If we are talking about a body of finite size, then first it is mentally broken into small parts. Thus, the system of material points is considered again, but its momentum is calculated not by ordinary summation, but by integration (see formula 4).
As you can see, there is no time dependence, therefore, the impulse of the system, which is not affected by external forces (or their influence is mutually compensated), remains unchanged over time.
Proof of the conservation law
Let's continue to consider a body of finite size as a system of material points. For each of them, Newton's Second Law is formulated according to Formula 5.
Let's pay attention to the fact that the system is closed. Then, summing over all points and applying Newton's Third Law, we obtain expression 6.
Thus, the impulse of a closed system is constant.
The conservation law is also valid in those cases when the total amount of forces that act on the system from the outside is equal to zero. One important particular statement follows from this. It says that the impulse of the body is constant if there is no external influence or the influence of several forces is compensated. For example, in the absence of friction after hitting with a stick, the puck must retain its momentum. Such a situation will be observed even in spite of the fact that the body is affected by the force of gravity and the reaction of the support (ice), since, although they are equal in magnitude, they are directed in opposite directions, i.e., they compensate each other.
Properties
The momentum of a body or material point is an additive quantity. What does it mean? Everything is simple: the impulse of a mechanical system of material points consists of the impulses of all material points included in the system.
The second property of this quantity is that it remains unchanged during interactions that change only the mechanical characteristics of the system.
In addition, the momentum is invariant with respect to any rotation of the frame of reference.
Relativistic case
Suppose that we are talking about non-interacting material points with speeds of the order of 10 to the 8th power or slightly less in the SI system. The three-dimensional impulse is calculated by formula 7, where c is understood as the speed of light in a vacuum.
In the case when it is closed, the law of conservation of momentum is true. At the same time, the three-dimensional momentum is not a relativistically invariant quantity, since there is its dependence on the frame of reference. There is also a 4D option. For one material point, it is determined by formula 8.
Impulse and energy
These quantities, as well as mass, are closely related to each other. In practical problems, relations (9) and (10) are usually used.
Definition through de Broglie waves
In 1924, it was hypothesized that not only photons, but also any other particles (protons, electrons, atoms) possess wave-particle duality. Its author was the French scientist Louis de Broglie. If we translate this hypothesis into the language of mathematics, then we can assert that with any particle that has energy and momentum, a wave is associated with a frequency and length expressed by formulas 11 and 12, respectively (h is Planck's constant).
From the last relation, we find that the pulse modulus and the wavelength denoted by the letter "lambda" are inversely proportional to each other (13).
If a particle with a relatively low energy is considered, which moves at a speed incommensurate with the speed of light, then the modulus of the momentum is calculated in the same way as in classical mechanics (see formula 1). Therefore, the wavelength is calculated according to expression 14. In other words, it is inversely proportional to the product of the mass and velocity of the particle, ie, its momentum.
Now you know that the impulse of a body is a measure of mechanical movement, and you have become familiar with its properties. Among them, in practical terms, the Law of Conservation is especially important. Even people far from physics observe it in everyday life. For example, everyone knows that firearms and artillery pieces give recoil when fired. The law of conservation of momentum is clearly demonstrated by the game of billiards. With its help, you can predict the direction of expansion of the balls after impact.
The law has found application in the calculations necessary to study the consequences of possible explosions, in the creation of jet vehicles, in the design of firearms and in many other areas of life.
A .22 caliber bullet has a mass of only 2 g. If you throw such a bullet at someone, he can easily catch it even without gloves. If you try to catch such a bullet that flew out of the muzzle at a speed of 300 m / s, then even gloves will not help here.
If a toy cart rolls on you, you can stop it with your toe. If a truck rolls on you, you should get out of the way.
Consider a problem that demonstrates the relationship between the impulse of force and the change in impulse of the body.
Example. The mass of the ball is 400 g, the speed that the ball acquired after impact is 30 m / s. The force with which the leg acted on the ball was 1500 N, and the impact time was 8 ms. Find the momentum of the force and the change in momentum of the body for the ball.
Body impulse change
Example. Estimate the average force from the floor on the ball during the kick.
1) During the impact, two forces act on the ball: the reaction force of the support, the force of gravity.
The reaction force changes over the time of the impact, so it is possible to find the average sex reaction force.
2) Change of momentum 
body shown in the figure
3) From Newton's second law
The main thing to remember
1) Formulas of body impulse, impulse of force;
2) Direction of the impulse vector;
3) Find the change in momentum of the body
General derivation of Newton's second law
Graph F (t). Variable strength
The impulse of force is numerically equal to the area of the figure under the F (t) graph.
If the force is not constant in time, for example, it increases linearly F = kt, then the impulse of this force is equal to the area of the triangle. You can replace this force with such a constant force that will change the momentum of the body by the same amount over the same period of time.
Average resultant force
THE LAW OF CONSERVATION OF IMPULSE
Online testing
Closed system of bodies
It is a system of bodies that only interact with each other. There are no external forces of interaction.
In the real world, such a system cannot exist; there is no way to remove all external interaction. A closed system of bodies is a physical model, just like a material point is a model. This is a model of a system of bodies that supposedly interact only with each other, external forces are not taken into account, they are neglected.
Momentum conservation law
In a closed system of bodies vector the sum of impulses of bodies does not change when bodies interact. If the impulse of one body has increased, this means that the impulse of some other body (or several bodies) at that moment has decreased by exactly the same amount.
Let's consider an example. The girl and the boy are skating. A closed system of bodies - a girl and a boy (we neglect friction and other external forces). The girl stands still, her momentum is zero, since the speed is zero (see the formula for the momentum of the body). After the boy, moving at a certain speed, collides with the girl, she will also begin to move. Now her body has impulse. The numerical value of the girl's impulse is exactly the same as by how much the boy's impulse decreased after the collision.
One body weighing 20 kg moves at a speed, the second body weighing 4 kg moves in the same direction at a speed. What are the impulses of each body. What is the momentum of the system?
The impulse of the system of bodies is the vector sum of the impulses of all bodies included in the system. In our example, this is the sum of two vectors (since we are considering two bodies), which are directed in the same direction, therefore
Now let's calculate the momentum of the system of bodies from the previous example, if the second body moves in the opposite direction.
Since the bodies move in opposite directions, we get the vector sum of impulses in different directions. More on the sum of vectors.
The main thing to remember
1) What is a closed system of bodies;
2) The law of conservation of momentum and its application
Impulse in physics
In translation from Latin "impulse" means "push". This physical quantity also called "amount of motion". It was introduced into science at about the same time as Newton's laws were discovered (at the end of the 17th century).
The branch of physics that studies the movement and interaction of material bodies is mechanics. An impulse in mechanics is a vector quantity equal to the product of a body's mass by its velocity: p = mv. The directions of the momentum and velocity vectors always coincide.
In the SI system, the unit of impulse is taken to be the impulse of a body weighing 1 kg, which moves at a speed of 1 m / s. Therefore, the SI unit of momentum is 1 kg ∙ m / s.
In computational problems, the projections of the velocity and momentum vectors on any axis are considered and equations for these projections are used: for example, if the x axis is selected, then the projections v (x) and p (x) are considered. By definition of momentum, these quantities are related by the relationship: p (x) = mv (x).
Depending on which axis is selected and where it is directed, the projection of the impulse vector onto it can be either positive or negative.
Momentum conservation law
The impulses of material bodies during their physical interaction can change. For example, when two balls, suspended on threads, collide, their impulses mutually change: one ball can move from a stationary state or increase its speed, while the other, on the contrary, can decrease its speed or stop. However, in a closed system, i.e. when the bodies interact only with each other and are not subjected to the influence of external forces, the vector sum of the impulses of these bodies remains constant for any of their interactions and movements. This is the law of conservation of momentum. Mathematically, it can be deduced from Newton's laws.
The law of conservation of momentum is also applicable to such systems where some external forces act on bodies, but their vector sum is equal to zero (for example, the force of gravity is balanced by the force of elasticity of the surface). Conventionally, such a system can also be considered closed.
In mathematical form, the law of conservation of momentum is written as follows: p1 + p2 +… + p (n) = p1 ’+ p2’ +… + p (n) ’(momenta p are vectors). For a two-body system, this equation looks like p1 + p2 = p1 ’+ p2’, or m1v1 + m2v2 = m1v1 ’+ m2v2’. For example, in the considered case with balls, the total momentum of both balls before interaction will be equal to the total momentum after interaction.
